Let $ \overline {\mathcal{M}_{g_{i}, n_{i}}}, \ i \in \{1, 2\},$ be a moduli stack of pointed stable curves of type $ (g_{i}, n_{i})$ over a finite field $ \mathbb{F}_{p}$ . For any algebraic stack $ \mathcal{X}$ , we write $ |\mathcal{X}|$ for the underlying topological space of $ \mathcal{X}$ . My question is as follows:

Is the natural surjective morphism of the underlying topological spaces $ $ |\overline {\mathcal{M_{g_{1}, n_{1}}}}\times_{\mathbb{F}_{p}}\overline {\mathcal{M_{g_{2}, n_{2}}}}| \rightarrow |\overline {\mathcal{M_{g_{1}, n_{1}}}}|\times_{|\text{Spec} \ \mathbb{F}_{p}|}|\overline {\mathcal{M_{g_{2}, n_{2}}}}|$ $ a homeomorphism?

I know that the morphism between underlying topological space of the fiber product of two algebraic stacks and the fiber product of the underlying topological space of two algebraic stacks is not a homeomorphism in general. Does the special case hold?